Solve for $x$ : $ 5|x + 9| - 4 = -3|x + 9| + 2 $
Add $ {3|x + 9|} $ to both sides: $ \begin{eqnarray} 5|x + 9| - 4 &=& -3|x + 9| + 2 \\ \\ { + 3|x + 9|} && { + 3|x + 9|} \\ \\ 8|x + 9| - 4 &=& 2 \end{eqnarray} $ Add ${4}$ to both sides: $ \begin{eqnarray} 8|x + 9| - 4 &=& 2 \\ \\ { + 4} &=& { + 4} \\ \\ 8|x + 9| &=& 6 \end{eqnarray} $ Divide both sides by ${8}$ $ \dfrac{8|x + 9|} {{8}} = \dfrac{6} {{8}} $ Simplify: $ |x + 9| = \dfrac{3}{4}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 9 = -\dfrac{3}{4} $ or $ x + 9 = \dfrac{3}{4} $ Solve for the solution where $x + 9$ is negative: $ x + 9 = -\dfrac{3}{4} $ Subtract ${9}$ from both sides: $ \begin{eqnarray} x + 9 &=& -\dfrac{3}{4} \\ \\ {- 9} && {- 9} \\ \\ x &=& -\dfrac{3}{4} - 9 \end{eqnarray} $ Change the ${ - 9}$ to an equivalent fraction with a denominator of $4$ $ x = - \dfrac{3}{4} {- \dfrac{36}{4}} $ $ x = -\dfrac{39}{4} $ Then calculate the solution where $x + 9$ is positive: $ x + 9 = \dfrac{3}{4} $ Subtract ${9}$ from both sides: $ \begin{eqnarray} x + 9 &=& \dfrac{3}{4} \\ \\ {- 9} && {- 9} \\ \\ x &=& \dfrac{3}{4} - 9 \end{eqnarray} $ Change the ${ - 9}$ to an equivalent fraction with a denominator of $4$ $ x = \dfrac{3}{4} {- \dfrac{36}{4}} $ $ x = -\dfrac{33}{4} $ Thus, the correct answer is $x = -\dfrac{39}{4} $ or $x = -\dfrac{33}{4} $.